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28       XII   TOPOLOGY AND TOPOLOGICAL ALGEBRA

(12.7.11)   For every sequence (jcn) in R, lim inf xn (resp. lim sup xn) is the least

H-00                                W~*00

(resp. the greatest) of the cluster values of the sequence (xn).

Let a = lim inf xn , and suppose first that a is finite. If c < a < c' and if m is

n-+oo

any integer, there exists n > m such that c < inf xn+p < c', hence there exists

p^O

p ^ 0 such that c < xn+p < c'. This shows that a is a cluster value of the
sequence (xn) (3.13.11). If fl~ +00 (resp. a=  oo), the argument is the
same if we replace c' (resp. c) by a. Conversely, if b is a cluster value of (xn)9
then it is also a cluster value of every sequence (xn+p)p^Q9 and therefore
b  infxn+p (3.13.7). Hence b ^ a.

p0

This result shows that the existence of a limit- of the sequence (xn) is
equivalent to the relation lim inf xn = lim sup xn 9 and that the common value

tt-00                        7I-+00

of the two sides is then the limit of (xn) (3.16.4).

It follows from the definition that if (xni) is any subsequence of a sequence
(xn) in R, then we have

(1 2.7.1 2)                          lim inf xnk ^ lim inf xn .

fc-oo                   -n-oo

(12.7.13) Let E be a Hausdorff topological space and let f be a mapping ofE
into R which is lower semicontinuous at a point aeE. Then for every sequence
of points ofE such that lim xn == a, we have

(12.7.13.1)                          lim inf /(*) ^ /(a).

H-+00

For every a <f(d) there exists a neighborhood V of a in E such that
f(y) ^ a for all y e V; but there exists n0 such that xn e V for all n^n0.
Hence f(xn) ^ a for all n^n0) which proves the result.

PROBLEMS

1. Let E be a topological space. For each closed subset A of the product space E x ft,
show that the mapping #h->inf (A(x)) ;of priA into ft is lower semicontinuous.
Conversely, if/: E ~>ft is lower semicontinuous, then the subset B of E x ft consisting
of all pairs (x, y) such that f(x) < y is closed in E x ft.  M-*OO \JpisO             /                   W-+00                  n-*cxD Vp^O              /